LMFDB - Embedded newform 415.4.a.c.1.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,4,Mod(1,415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("415.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	415.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.4857926524$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	415.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.28686 q^{2} -0.965562 q^{3} +10.3772 q^{4} -5.00000 q^{5} -4.13923 q^{6} +6.97800 q^{7} +10.1906 q^{8} -26.0677 q^{9} -21.4343 q^{10} -52.3762 q^{11} -10.0198 q^{12} -27.1779 q^{13} +29.9137 q^{14} +4.82781 q^{15} -39.3317 q^{16} -4.95837 q^{17} -111.749 q^{18} +105.160 q^{19} -51.8858 q^{20} -6.73769 q^{21} -224.530 q^{22} -160.148 q^{23} -9.83965 q^{24} +25.0000 q^{25} -116.508 q^{26} +51.2401 q^{27} +72.4119 q^{28} -94.5851 q^{29} +20.6961 q^{30} +4.25766 q^{31} -250.134 q^{32} +50.5725 q^{33} -21.2558 q^{34} -34.8900 q^{35} -270.509 q^{36} +396.186 q^{37} +450.807 q^{38} +26.2419 q^{39} -50.9530 q^{40} -122.063 q^{41} -28.8835 q^{42} -111.330 q^{43} -543.517 q^{44} +130.338 q^{45} -686.532 q^{46} +238.641 q^{47} +37.9772 q^{48} -294.307 q^{49} +107.172 q^{50} +4.78761 q^{51} -282.029 q^{52} -187.155 q^{53} +219.659 q^{54} +261.881 q^{55} +71.1100 q^{56} -101.539 q^{57} -405.473 q^{58} -231.622 q^{59} +50.0990 q^{60} +5.25376 q^{61} +18.2520 q^{62} -181.900 q^{63} -757.637 q^{64} +135.889 q^{65} +216.797 q^{66} -336.053 q^{67} -51.4539 q^{68} +154.633 q^{69} -149.569 q^{70} -705.184 q^{71} -265.645 q^{72} +675.753 q^{73} +1698.39 q^{74} -24.1390 q^{75} +1091.27 q^{76} -365.482 q^{77} +112.495 q^{78} +512.587 q^{79} +196.659 q^{80} +654.352 q^{81} -523.267 q^{82} +83.0000 q^{83} -69.9182 q^{84} +24.7919 q^{85} -477.254 q^{86} +91.3277 q^{87} -533.745 q^{88} -123.522 q^{89} +558.743 q^{90} -189.647 q^{91} -1661.88 q^{92} -4.11103 q^{93} +1023.02 q^{94} -525.801 q^{95} +241.520 q^{96} -302.081 q^{97} -1261.65 q^{98} +1365.33 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 5 q^{2} - 12 q^{3} + 87 q^{4} - 105 q^{5} - 7 q^{6} - 11 q^{7} - 84 q^{8} + 153 q^{9} + 25 q^{10} - 30 q^{11} - 244 q^{12} - 89 q^{13} - 191 q^{14} + 60 q^{15} + 583 q^{16} - 357 q^{17} - 281 q^{18}+ \cdots - 5369 q^{99}+O(q^{100}) $ 21 * q - 5 * q^2 - 12 * q^3 + 87 * q^4 - 105 * q^5 - 7 * q^6 - 11 * q^7 - 84 * q^8 + 153 * q^9 + 25 * q^10 - 30 * q^11 - 244 * q^12 - 89 * q^13 - 191 * q^14 + 60 * q^15 + 583 * q^16 - 357 * q^17 - 281 * q^18 - 175 * q^19 - 435 * q^20 - 41 * q^21 - 122 * q^22 - 554 * q^23 - 136 * q^24 + 525 * q^25 - 933 * q^26 - 555 * q^27 - 773 * q^28 - 96 * q^29 + 35 * q^30 + 99 * q^31 - 1307 * q^32 - 1097 * q^33 + 393 * q^34 + 55 * q^35 - 746 * q^36 - 756 * q^37 - 1192 * q^38 + 40 * q^39 + 420 * q^40 - 1436 * q^41 - 1112 * q^42 - 1003 * q^43 - 706 * q^44 - 765 * q^45 - 775 * q^46 - 858 * q^47 - 2837 * q^48 + 1384 * q^49 - 125 * q^50 - 279 * q^51 - 1092 * q^52 - 1625 * q^53 - 269 * q^54 + 150 * q^55 - 3806 * q^56 - 1740 * q^57 - 1087 * q^58 - 1224 * q^59 + 1220 * q^60 - 794 * q^61 - 2979 * q^62 - 1236 * q^63 + 2570 * q^64 + 445 * q^65 - 1094 * q^66 - 571 * q^67 - 4064 * q^68 + 774 * q^69 + 955 * q^70 - 1078 * q^71 - 2610 * q^72 - 1986 * q^73 - 1872 * q^74 - 300 * q^75 - 2854 * q^76 - 2519 * q^77 - 2449 * q^78 - 432 * q^79 - 2915 * q^80 - 219 * q^81 - 2143 * q^82 + 1743 * q^83 - 377 * q^84 + 1785 * q^85 + 1556 * q^86 - 5681 * q^87 - 3158 * q^88 - 6252 * q^89 + 1405 * q^90 + 684 * q^91 - 7623 * q^92 - 3314 * q^93 - 3877 * q^94 + 875 * q^95 - 954 * q^96 - 4880 * q^97 - 5072 * q^98 - 5369 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.28686	1.51563	0.757817	−	0.652467i	$-0.226266\pi$
$2$	4.28686	1.51563	0.757817	+	0.652467i	$0.226266\pi$
$3$	−0.965562	−0.185822	−0.0929112	−	0.995674i	$-0.529617\pi$
$3$	−0.965562	−0.185822	−0.0929112	+	0.995674i	$0.529617\pi$
$4$	10.3772	1.29715
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−4.13923	−0.281639
$7$	6.97800	0.376777	0.188388	−	0.982095i	$-0.439674\pi$
$7$	6.97800	0.376777	0.188388	+	0.982095i	$0.439674\pi$
$8$	10.1906	0.450365
$9$	−26.0677	−0.965470
$10$	−21.4343	−0.677812
$11$	−52.3762	−1.43564	−0.717820	−	0.696229i	$-0.754860\pi$
$11$	−52.3762	−1.43564	−0.717820	+	0.696229i	$0.754860\pi$
$12$	−10.0198	−0.241039
$13$	−27.1779	−0.579829	−0.289915	−	0.957052i	$-0.593627\pi$
$13$	−27.1779	−0.579829	−0.289915	+	0.957052i	$0.593627\pi$
$14$	29.9137	0.571056
$15$	4.82781	0.0831023
$16$	−39.3317	−0.614558
$17$	−4.95837	−0.0707401	−0.0353701	−	0.999374i	$-0.511261\pi$
$17$	−4.95837	−0.0707401	−0.0353701	+	0.999374i	$0.511261\pi$
$18$	−111.749	−1.46330
$19$	105.160	1.26976	0.634880	−	0.772611i	$-0.281050\pi$
$19$	105.160	1.26976	0.634880	+	0.772611i	$0.281050\pi$
$20$	−51.8858	−0.580101
$21$	−6.73769	−0.0700136
$22$	−224.530	−2.17590
$23$	−160.148	−1.45188	−0.725938	−	0.687760i	$-0.758594\pi$
$23$	−160.148	−1.45188	−0.725938	+	0.687760i	$0.758594\pi$
$24$	−9.83965	−0.0836879
$25$	25.0000	0.200000
$26$	−116.508	−0.878809
$27$	51.2401	0.365228
$28$	72.4119	0.488735
$29$	−94.5851	−0.605656	−0.302828	−	0.953045i	$-0.597931\pi$
$29$	−94.5851	−0.605656	−0.302828	+	0.953045i	$0.597931\pi$
$30$	20.6961	0.125953
$31$	4.25766	0.0246677	0.0123338	−	0.999924i	$-0.496074\pi$
$31$	4.25766	0.0246677	0.0123338	+	0.999924i	$0.496074\pi$
$32$	−250.134	−1.38181
$33$	50.5725	0.266774
$34$	−21.2558	−0.107216
$35$	−34.8900	−0.168500
$36$	−270.509	−1.25236
$37$	396.186	1.76034	0.880170	−	0.474659i	$-0.157429\pi$
$37$	396.186	1.76034	0.880170	+	0.474659i	$0.157429\pi$
$38$	450.807	1.92449
$39$	26.2419	0.107745
$40$	−50.9530	−0.201409
$41$	−122.063	−0.464952	−0.232476	−	0.972602i	$-0.574683\pi$
$41$	−122.063	−0.464952	−0.232476	+	0.972602i	$0.574683\pi$
$42$	−28.8835	−0.106115
$43$	−111.330	−0.394828	−0.197414	−	0.980320i	$-0.563254\pi$
$43$	−111.330	−0.394828	−0.197414	+	0.980320i	$0.563254\pi$
$44$	−543.517	−1.86223
$45$	130.338	0.431771
$46$	−686.532	−2.20051
$47$	238.641	0.740623	0.370312	−	0.928908i	$-0.379251\pi$
$47$	238.641	0.740623	0.370312	+	0.928908i	$0.379251\pi$
$48$	37.9772	0.114199
$49$	−294.307	−0.858039
$50$	107.172	0.303127
$51$	4.78761	0.0131451
$52$	−282.029	−0.752123
$53$	−187.155	−0.485050	−0.242525	−	0.970145i	$-0.577976\pi$
$53$	−187.155	−0.485050	−0.242525	+	0.970145i	$0.577976\pi$
$54$	219.659	0.553553
$55$	261.881	0.642037
$56$	71.1100	0.169687
$57$	−101.539	−0.235950
$58$	−405.473	−0.917952
$59$	−231.622	−0.511096	−0.255548	−	0.966796i	$-0.582256\pi$
$59$	−231.622	−0.511096	−0.255548	+	0.966796i	$0.582256\pi$
$60$	50.0990	0.107796
$61$	5.25376	0.0110275	0.00551373	−	0.999985i	$-0.498245\pi$
$61$	5.25376	0.0110275	0.00551373	+	0.999985i	$0.498245\pi$
$62$	18.2520	0.0373872
$63$	−181.900	−0.363767
$64$	−757.637	−1.47976
$65$	135.889	0.259308
$66$	216.797	0.404332
$67$	−336.053	−0.612768	−0.306384	−	0.951908i	$-0.599119\pi$
$67$	−336.053	−0.612768	−0.306384	+	0.951908i	$0.599119\pi$
$68$	−51.4539	−0.0917603
$69$	154.633	0.269791
$70$	−149.569	−0.255384
$71$	−705.184	−1.17873	−0.589366	−	0.807866i	$-0.700622\pi$
$71$	−705.184	−1.17873	−0.589366	+	0.807866i	$0.700622\pi$
$72$	−265.645	−0.434814
$73$	675.753	1.08344	0.541718	−	0.840560i	$-0.317774\pi$
$73$	675.753	1.08344	0.541718	+	0.840560i	$0.317774\pi$
$74$	1698.39	2.66803
$75$	−24.1390	−0.0371645
$76$	1091.27	1.64706
$77$	−365.482	−0.540915
$78$	112.495	0.163302
$79$	512.587	0.730007	0.365003	−	0.931006i	$-0.381068\pi$
$79$	512.587	0.730007	0.365003	+	0.931006i	$0.381068\pi$
$80$	196.659	0.274839
$81$	654.352	0.897602
$82$	−523.267	−0.704698
$83$	83.0000	0.109764
$83$	83.0000	0.109764
$84$	−69.9182	−0.0908178
$85$	24.7919	0.0316359
$86$	−477.254	−0.598414
$87$	91.3277	0.112544
$88$	−533.745	−0.646561
$89$	−123.522	−0.147116	−0.0735578	−	0.997291i	$-0.523435\pi$
$89$	−123.522	−0.147116	−0.0735578	+	0.997291i	$0.523435\pi$
$90$	558.743	0.654407
$91$	−189.647	−0.218466
$92$	−1661.88	−1.88330
$93$	−4.11103	−0.00458381
$94$	1023.02	1.12251
$95$	−525.801	−0.567854
$96$	241.520	0.256771
$97$	−302.081	−0.316203	−0.158102	−	0.987423i	$-0.550537\pi$
$97$	−302.081	−0.316203	−0.158102	+	0.987423i	$0.550537\pi$
$98$	−1261.65	−1.30047
$99$	1365.33	1.38607
$100$	259.429	0.259429
$101$	−951.443	−0.937348	−0.468674	−	0.883371i	$-0.655268\pi$
$101$	−951.443	−0.937348	−0.468674	+	0.883371i	$0.655268\pi$
$102$	20.5238	0.0199232
$103$	969.995	0.927927	0.463963	−	0.885854i	$-0.346427\pi$
$103$	969.995	0.927927	0.463963	+	0.885854i	$0.346427\pi$
$104$	−276.958	−0.261135
$105$	33.6885	0.0313110
$106$	−802.306	−0.735159
$107$	−347.744	−0.314184	−0.157092	−	0.987584i	$-0.550212\pi$
$107$	−347.744	−0.314184	−0.157092	+	0.987584i	$0.550212\pi$
$108$	531.727	0.473755
$109$	341.509	0.300098	0.150049	−	0.988679i	$-0.452057\pi$
$109$	341.509	0.300098	0.150049	+	0.988679i	$0.452057\pi$
$110$	1122.65	0.973094
$111$	−382.542	−0.327111
$112$	−274.457	−0.231551
$113$	1701.22	1.41626	0.708131	−	0.706081i	$-0.249539\pi$
$113$	1701.22	1.41626	0.708131	+	0.706081i	$0.249539\pi$
$114$	−435.282	−0.357613
$115$	800.740	0.649299
$116$	−981.526	−0.785624
$117$	708.464	0.559808
$118$	−992.933	−0.774635
$119$	−34.5995	−0.0266532
$120$	49.1982	0.0374264
$121$	1412.27	1.06106
$122$	22.5221	0.0167136
$123$	117.859	0.0863986
$124$	44.1825	0.0319976
$125$	−125.000	−0.0894427
$126$	−779.782	−0.551337
$127$	−2285.71	−1.59704	−0.798521	−	0.601967i	$-0.794384\pi$
$127$	−2285.71	−1.59704	−0.798521	+	0.601967i	$0.794384\pi$
$128$	−1246.81	−0.860964
$129$	107.496	0.0733678
$130$	582.538	0.393015
$131$	−1334.11	−0.889784	−0.444892	−	0.895584i	$-0.646758\pi$
$131$	−1334.11	−0.889784	−0.444892	+	0.895584i	$0.646758\pi$
$132$	524.799	0.346045
$133$	733.809	0.478416
$134$	−1440.61	−0.928732
$135$	−256.201	−0.163335
$136$	−50.5288	−0.0318589
$137$	997.257	0.621908	0.310954	−	0.950425i	$-0.399351\pi$
$137$	997.257	0.621908	0.310954	+	0.950425i	$0.399351\pi$
$138$	662.889	0.408905
$139$	2365.08	1.44319	0.721595	−	0.692316i	$-0.243409\pi$
$139$	2365.08	1.44319	0.721595	+	0.692316i	$0.243409\pi$
$140$	−362.060	−0.218569
$141$	−230.422	−0.137624
$142$	−3023.03	−1.78653
$143$	1423.47	0.832426
$144$	1025.29	0.593337
$145$	472.926	0.270857
$146$	2896.86	1.64209
$147$	284.172	0.159443
$148$	4111.29	2.28342
$149$	36.4519	0.0200420	0.0100210	−	0.999950i	$-0.496810\pi$
$149$	36.4519	0.0200420	0.0100210	+	0.999950i	$0.496810\pi$
$150$	−103.481	−0.0563278
$151$	2527.30	1.36204	0.681022	−	0.732263i	$-0.261536\pi$
$151$	2527.30	1.36204	0.681022	+	0.732263i	$0.261536\pi$
$152$	1071.65	0.571855
$153$	129.253	0.0682975
$154$	−1566.77	−0.819830
$155$	−21.2883	−0.0110317
$156$	272.317	0.139761
$157$	779.620	0.396309	0.198154	−	0.980171i	$-0.436505\pi$
$157$	779.620	0.396309	0.198154	+	0.980171i	$0.436505\pi$
$158$	2197.39	1.10642
$159$	180.709	0.0901332
$160$	1250.67	0.617964
$161$	−1117.51	−0.547033
$162$	2805.12	1.36044
$163$	−726.658	−0.349179	−0.174590	−	0.984641i	$-0.555860\pi$
$163$	−726.658	−0.349179	−0.174590	+	0.984641i	$0.555860\pi$
$164$	−1266.67	−0.603111
$165$	−252.862	−0.119305
$166$	355.809	0.166362
$167$	−901.997	−0.417956	−0.208978	−	0.977920i	$-0.567014\pi$
$167$	−901.997	−0.417956	−0.208978	+	0.977920i	$0.567014\pi$
$168$	−68.6611	−0.0315317
$169$	−1458.36	−0.663798
$170$	106.279	0.0479485
$171$	−2741.29	−1.22591
$172$	−1155.29	−0.512149
$173$	991.536	0.435752	0.217876	−	0.975976i	$-0.430087\pi$
$173$	991.536	0.435752	0.217876	+	0.975976i	$0.430087\pi$
$174$	391.509	0.170576
$175$	174.450	0.0753554
$176$	2060.05	0.882283
$177$	223.646	0.0949731
$178$	−529.521	−0.222973
$179$	−2921.59	−1.21995	−0.609973	−	0.792422i	$-0.708820\pi$
$179$	−2921.59	−1.21995	−0.609973	+	0.792422i	$0.708820\pi$
$180$	1352.54	0.560071
$181$	−2746.34	−1.12781	−0.563905	−	0.825840i	$-0.690702\pi$
$181$	−2746.34	−1.12781	−0.563905	+	0.825840i	$0.690702\pi$
$182$	−812.991	−0.331115
$183$	−5.07283	−0.00204915
$184$	−1632.00	−0.653874
$185$	−1980.93	−0.787248
$186$	−17.6234	−0.00694738
$187$	259.701	0.101557
$188$	2476.41	0.960697
$189$	357.554	0.137610
$190$	−2254.04	−0.860658
$191$	1926.08	0.729667	0.364833	−	0.931073i	$-0.381126\pi$
$191$	1926.08	0.729667	0.364833	+	0.931073i	$0.381126\pi$
$192$	731.545	0.274973
$193$	−1638.01	−0.610916	−0.305458	−	0.952206i	$-0.598810\pi$
$193$	−1638.01	−0.610916	−0.305458	+	0.952206i	$0.598810\pi$
$194$	−1294.98	−0.479249
$195$	−131.209	−0.0481852
$196$	−3054.08	−1.11300
$197$	−5382.81	−1.94675	−0.973374	−	0.229222i	$-0.926382\pi$
$197$	−5382.81	−1.94675	−0.973374	+	0.229222i	$0.926382\pi$
$198$	5852.97	2.10077
$199$	−1051.48	−0.374560	−0.187280	−	0.982307i	$-0.559967\pi$
$199$	−1051.48	−0.374560	−0.187280	+	0.982307i	$0.559967\pi$
$200$	254.765	0.0900730
$201$	324.480	0.113866
$202$	−4078.71	−1.42068
$203$	−660.015	−0.228197
$204$	49.6819	0.0170511
$205$	610.315	0.207933
$206$	4158.23	1.40640
$207$	4174.69	1.40174
$208$	1068.95	0.356339
$209$	−5507.90	−1.82292
$210$	144.418	0.0474560
$211$	−1207.30	−0.393906	−0.196953	−	0.980413i	$-0.563105\pi$
$211$	−1207.30	−0.393906	−0.196953	+	0.980413i	$0.563105\pi$
$212$	−1942.14	−0.629181
$213$	680.899	0.219035
$214$	−1490.73	−0.476188
$215$	556.648	0.176572
$216$	522.167	0.164486
$217$	29.7100	0.00929421
$218$	1464.00	0.454839
$219$	−652.481	−0.201327
$220$	2717.59	0.832816
$221$	134.758	0.0410172
$222$	−1639.90	−0.495780
$223$	−2398.85	−0.720353	−0.360177	−	0.932884i	$-0.617284\pi$
$223$	−2398.85	−0.720353	−0.360177	+	0.932884i	$0.617284\pi$
$224$	−1745.44	−0.520634
$225$	−651.692	−0.193094
$226$	7292.90	2.14653
$227$	−2963.53	−0.866505	−0.433252	−	0.901273i	$-0.642634\pi$
$227$	−2963.53	−0.866505	−0.433252	+	0.901273i	$0.642634\pi$
$228$	−1053.68	−0.306061
$229$	−2097.05	−0.605139	−0.302569	−	0.953127i	$-0.597844\pi$
$229$	−2097.05	−0.605139	−0.302569	+	0.953127i	$0.597844\pi$
$230$	3432.66	0.984099
$231$	352.895	0.100514
$232$	−963.878	−0.272766
$233$	−1029.34	−0.289418	−0.144709	−	0.989474i	$-0.546225\pi$
$233$	−1029.34	−0.289418	−0.144709	+	0.989474i	$0.546225\pi$
$234$	3037.09	0.848464
$235$	−1193.20	−0.331217
$236$	−2403.59	−0.662966
$237$	−494.934	−0.135652
$238$	−148.323	−0.0403965
$239$	4082.53	1.10492	0.552462	−	0.833538i	$-0.313688\pi$
$239$	4082.53	1.10492	0.552462	+	0.833538i	$0.313688\pi$
$240$	−189.886	−0.0510712
$241$	−178.223	−0.0476364	−0.0238182	−	0.999716i	$-0.507582\pi$
$241$	−178.223	−0.0476364	−0.0238182	+	0.999716i	$0.507582\pi$
$242$	6054.21	1.60818
$243$	−2015.30	−0.532023
$244$	54.5192	0.0143042
$245$	1471.54	0.383727
$246$	505.247	0.130949
$247$	−2858.03	−0.736244
$248$	43.3881	0.0111095
$249$	−80.1416	−0.0203967
$250$	−535.858	−0.135562
$251$	0.912359	0.000229433 0	0.000114716	−	1.00000i	$-0.499963\pi$
$251$	0.912359	0.000229433 0	0.000114716		1.00000i	$0.499963\pi$
$252$	−1887.61	−0.471859
$253$	8387.95	2.08437
$254$	−9798.54	−2.42053
$255$	−23.9381	−0.00587867
$256$	716.198	0.174853
$257$	1726.48	0.419046	0.209523	−	0.977804i	$-0.432809\pi$
$257$	1726.48	0.419046	0.209523	+	0.977804i	$0.432809\pi$
$258$	460.818	0.111199
$259$	2764.59	0.663255
$260$	1410.15	0.336360
$261$	2465.62	0.584742
$262$	−5719.14	−1.34859
$263$	−2823.34	−0.661957	−0.330978	−	0.943638i	$-0.607379\pi$
$263$	−2823.34	−0.661957	−0.330978	+	0.943638i	$0.607379\pi$
$264$	515.364	0.120146
$265$	935.773	0.216921
$266$	3145.74	0.725103
$267$	119.268	0.0273374
$268$	−3487.28	−0.794850
$269$	−1169.54	−0.265086	−0.132543	−	0.991177i	$-0.542314\pi$
$269$	−1169.54	−0.265086	−0.132543	+	0.991177i	$0.542314\pi$
$270$	−1098.30	−0.247556
$271$	−5681.99	−1.27364	−0.636820	−	0.771013i	$-0.719750\pi$
$271$	−5681.99	−1.27364	−0.636820	+	0.771013i	$0.719750\pi$
$272$	195.021	0.0434739
$273$	183.116	0.0405959
$274$	4275.10	0.942585
$275$	−1309.41	−0.287128
$276$	1604.65	0.349959
$277$	2726.28	0.591359	0.295680	−	0.955287i	$-0.404454\pi$
$277$	2726.28	0.591359	0.295680	+	0.955287i	$0.404454\pi$
$278$	10138.8	2.18735
$279$	−110.987	−0.0238159
$280$	−355.550	−0.0758863
$281$	8522.15	1.80921	0.904606	−	0.426249i	$-0.140165\pi$
$281$	8522.15	1.80921	0.904606	+	0.426249i	$0.140165\pi$
$282$	−987.788	−0.208588
$283$	4143.06	0.870245	0.435122	−	0.900371i	$-0.356705\pi$
$283$	4143.06	0.870245	0.435122	+	0.900371i	$0.356705\pi$
$284$	−7317.82	−1.52899
$285$	507.694	0.105520
$286$	6102.23	1.26165
$287$	−851.757	−0.175183
$288$	6520.42	1.33410
$289$	−4888.41	−0.994996
$290$	2027.37	0.410521
$291$	291.678	0.0587577
$292$	7012.40	1.40538
$293$	8918.68	1.77828	0.889138	−	0.457639i	$-0.151305\pi$
$293$	8918.68	1.77828	0.889138	+	0.457639i	$0.151305\pi$
$294$	1218.21	0.241657
$295$	1158.11	0.228569
$296$	4037.37	0.792795
$297$	−2683.77	−0.524336
$298$	156.264	0.0303763
$299$	4352.48	0.841840
$300$	−250.495	−0.0482078
$301$	−776.858	−0.148762
$302$	10834.2	2.06436
$303$	918.677	0.174180
$304$	−4136.13	−0.780341
$305$	−26.2688	−0.00493163
$306$	554.091	0.103514
$307$	−4995.61	−0.928712	−0.464356	−	0.885649i	$-0.653714\pi$
$307$	−4995.61	−0.928712	−0.464356	+	0.885649i	$0.653714\pi$
$308$	−3792.66	−0.701646
$309$	−936.590	−0.172430
$310$	−91.2600	−0.0167201
$311$	−7955.03	−1.45044	−0.725222	−	0.688515i	$-0.758263\pi$
$311$	−7955.03	−1.45044	−0.725222	+	0.688515i	$0.758263\pi$
$312$	267.420	0.0485247
$313$	−8063.51	−1.45616	−0.728078	−	0.685495i	$-0.759586\pi$
$313$	−8063.51	−1.45616	−0.728078	+	0.685495i	$0.759586\pi$
$314$	3342.12	0.600659
$315$	909.502	0.162681
$316$	5319.20	0.946925
$317$	−6333.42	−1.12215	−0.561073	−	0.827766i	$-0.689611\pi$
$317$	−6333.42	−1.12215	−0.561073	+	0.827766i	$0.689611\pi$
$318$	774.676	0.136609
$319$	4954.01	0.869503
$320$	3788.18	0.661769
$321$	335.768	0.0583824
$322$	−4790.62	−0.829102
$323$	−521.424	−0.0898229
$324$	6790.32	1.16432
$325$	−679.446	−0.115966
$326$	−3115.08	−0.529228
$327$	−329.748	−0.0557649
$328$	−1243.90	−0.209398
$329$	1665.23	0.279050
$330$	−1083.99	−0.180823
$331$	5520.41	0.916704	0.458352	−	0.888771i	$-0.348440\pi$
$331$	5520.41	0.916704	0.458352	+	0.888771i	$0.348440\pi$
$332$	861.305	0.142380
$333$	−10327.7	−1.69956
$334$	−3866.74	−0.633468
$335$	1680.27	0.274038
$336$	265.005	0.0430274
$337$	−7886.38	−1.27477	−0.637386	−	0.770544i	$-0.719984\pi$
$337$	−7886.38	−1.27477	−0.637386	+	0.770544i	$0.719984\pi$
$338$	−6251.80	−1.00607
$339$	−1642.64	−0.263173
$340$	257.269	0.0410364
$341$	−223.000	−0.0354139
$342$	−11751.5	−1.85804
$343$	−4447.13	−0.700066
$344$	−1134.51	−0.177817
$345$	−773.163	−0.120654
$346$	4250.58	0.660440
$347$	9528.99	1.47419	0.737094	−	0.675790i	$-0.236198\pi$
$347$	9528.99	1.47419	0.737094	+	0.675790i	$0.236198\pi$
$348$	947.724	0.145987
$349$	1209.76	0.185549	0.0927747	−	0.995687i	$-0.470426\pi$
$349$	1209.76	0.185549	0.0927747	+	0.995687i	$0.470426\pi$
$350$	747.843	0.114211
$351$	−1392.60	−0.211770
$352$	13101.1	1.98378
$353$	−11162.7	−1.68309	−0.841544	−	0.540188i	$-0.818353\pi$
$353$	−11162.7	−1.68309	−0.841544	+	0.540188i	$0.818353\pi$
$354$	958.738	0.143944
$355$	3525.92	0.527145
$356$	−1281.81	−0.190830
$357$	33.4080	0.00495277
$358$	−12524.5	−1.84899
$359$	5231.99	0.769176	0.384588	−	0.923088i	$-0.374344\pi$
$359$	5231.99	0.769176	0.384588	+	0.923088i	$0.374344\pi$
$360$	1328.23	0.194455
$361$	4199.69	0.612289
$362$	−11773.2	−1.70935
$363$	−1363.63	−0.197169
$364$	−1968.00	−0.283383
$365$	−3378.76	−0.484527
$366$	−21.7465	−0.00310576
$367$	−2266.27	−0.322339	−0.161170	−	0.986927i	$-0.551527\pi$
$367$	−2266.27	−0.322339	−0.161170	+	0.986927i	$0.551527\pi$
$368$	6298.89	0.892262
$369$	3181.90	0.448898
$370$	−8491.97	−1.19318
$371$	−1305.97	−0.182756
$372$	−42.6609	−0.00594587
$373$	−14018.3	−1.94595	−0.972976	−	0.230905i	$-0.925831\pi$
$373$	−14018.3	−1.94595	−0.972976	+	0.230905i	$0.925831\pi$
$374$	1113.30	0.153924
$375$	120.695	0.0166205
$376$	2431.89	0.333551
$377$	2570.62	0.351177
$378$	1532.78	0.208566
$379$	−4261.85	−0.577617	−0.288808	−	0.957387i	$-0.593259\pi$
$379$	−4261.85	−0.577617	−0.288808	+	0.957387i	$0.593259\pi$
$380$	−5456.33	−0.736589
$381$	2207.00	0.296766
$382$	8256.84	1.10591
$383$	1332.68	0.177798	0.0888991	−	0.996041i	$-0.471665\pi$
$383$	1332.68	0.177798	0.0888991	+	0.996041i	$0.471665\pi$
$384$	1203.87	0.159986
$385$	1827.41	0.241905
$386$	−7021.94	−0.925925
$387$	2902.10	0.381194
$388$	−3134.75	−0.410162
$389$	12000.6	1.56416	0.782078	−	0.623180i	$-0.214160\pi$
$389$	12000.6	1.56416	0.782078	+	0.623180i	$0.214160\pi$
$390$	−562.477	−0.0730311
$391$	794.073	0.102706
$392$	−2999.17	−0.386431
$393$	1288.16	0.165342
$394$	−23075.4	−2.95056
$395$	−2562.93	−0.326469
$396$	14168.2	1.79793
$397$	−14026.0	−1.77316	−0.886579	−	0.462577i	$-0.846925\pi$
$397$	−14026.0	−1.77316	−0.886579	+	0.462577i	$0.846925\pi$
$398$	−4507.54	−0.567695
$399$	−708.538	−0.0889004
$400$	−983.293	−0.122912
$401$	−2185.21	−0.272130	−0.136065	−	0.990700i	$-0.543446\pi$
$401$	−2185.21	−0.272130	−0.136065	+	0.990700i	$0.543446\pi$
$402$	1391.00	0.172579
$403$	−115.714	−0.0143031
$404$	−9873.29	−1.21588
$405$	−3271.76	−0.401420
$406$	−2829.39	−0.345863
$407$	−20750.7	−2.52721
$408$	48.7886	0.00592009
$409$	−12102.4	−1.46314	−0.731569	−	0.681768i	$-0.761211\pi$
$409$	−12102.4	−1.46314	−0.731569	+	0.681768i	$0.761211\pi$
$410$	2616.34	0.315150
$411$	−962.913	−0.115564
$412$	10065.8	1.20366
$413$	−1616.26	−0.192569
$414$	17896.3	2.12453
$415$	−415.000	−0.0490881
$416$	6798.11	0.801214
$417$	−2283.63	−0.268177
$418$	−23611.6	−2.76287
$419$	6549.47	0.763634	0.381817	−	0.924238i	$-0.375298\pi$
$419$	6549.47	0.763634	0.381817	+	0.924238i	$0.375298\pi$
$420$	349.591	0.0406150
$421$	7319.82	0.847379	0.423689	−	0.905808i	$-0.360735\pi$
$421$	7319.82	0.847379	0.423689	+	0.905808i	$0.360735\pi$
$422$	−5175.54	−0.597018
$423$	−6220.81	−0.715050
$424$	−1907.22	−0.218450
$425$	−123.959	−0.0141480
$426$	2918.92	0.331977
$427$	36.6608	0.00415489
$428$	−3608.60	−0.407542
$429$	−1374.45	−0.154683
$430$	2386.27	0.267619
$431$	−15959.4	−1.78361	−0.891804	−	0.452422i	$-0.850560\pi$
$431$	−15959.4	−1.78361	−0.891804	+	0.452422i	$0.850560\pi$
$432$	−2015.36	−0.224454
$433$	−6991.45	−0.775953	−0.387977	−	0.921669i	$-0.626826\pi$
$433$	−6991.45	−0.775953	−0.387977	+	0.921669i	$0.626826\pi$
$434$	127.363	0.0140866
$435$	−456.639	−0.0503314
$436$	3543.90	0.389271
$437$	−16841.2	−1.84353
$438$	−2797.09	−0.305138
$439$	17104.6	1.85959	0.929793	−	0.368082i	$-0.119985\pi$
$439$	17104.6	1.85959	0.929793	+	0.368082i	$0.119985\pi$
$440$	2668.72	0.289151
$441$	7671.92	0.828411
$442$	577.688	0.0621670
$443$	−3203.62	−0.343586	−0.171793	−	0.985133i	$-0.554956\pi$
$443$	−3203.62	−0.343586	−0.171793	+	0.985133i	$0.554956\pi$
$444$	−3969.70	−0.424310
$445$	617.609	0.0657921
$446$	−10283.5	−1.09179
$447$	−35.1965	−0.00372425
$448$	−5286.79	−0.557539
$449$	−85.9418	−0.00903306	−0.00451653	−	0.999990i	$-0.501438\pi$
$449$	−85.9418	−0.00903306	−0.00451653	+	0.999990i	$0.501438\pi$
$450$	−2793.71	−0.292660
$451$	6393.21	0.667504
$452$	17653.9	1.83710
$453$	−2440.26	−0.253098
$454$	−12704.2	−1.31330
$455$	948.236	0.0977011
$456$	−1034.74	−0.106263
$457$	−4747.44	−0.485943	−0.242972	−	0.970033i	$-0.578122\pi$
$457$	−4747.44	−0.485943	−0.242972	+	0.970033i	$0.578122\pi$
$458$	−8989.74	−0.917168
$459$	−254.068	−0.0258363
$460$	8309.41	0.842235
$461$	1456.22	0.147121	0.0735606	−	0.997291i	$-0.476564\pi$
$461$	1456.22	0.147121	0.0735606	+	0.997291i	$0.476564\pi$
$462$	1512.81	0.152343
$463$	1387.55	0.139276	0.0696379	−	0.997572i	$-0.477816\pi$
$463$	1387.55	0.139276	0.0696379	+	0.997572i	$0.477816\pi$
$464$	3720.19	0.372210
$465$	20.5552	0.00204994
$466$	−4412.64	−0.438651
$467$	19129.3	1.89550	0.947752	−	0.319009i	$-0.103350\pi$
$467$	19129.3	1.89550	0.947752	+	0.319009i	$0.103350\pi$
$468$	7351.85	0.726153
$469$	−2344.98	−0.230877
$470$	−5115.09	−0.502003
$471$	−752.772	−0.0736431
$472$	−2360.37	−0.230180
$473$	5831.02	0.566830
$474$	−2121.71	−0.205598
$475$	2629.01	0.253952
$476$	−359.045	−0.0345731
$477$	4878.69	0.468302
$478$	17501.2	1.67466
$479$	−8270.27	−0.788890	−0.394445	−	0.918920i	$-0.629063\pi$
$479$	−8270.27	−0.788890	−0.394445	+	0.918920i	$0.629063\pi$
$480$	−1207.60	−0.114832
$481$	−10767.5	−1.02070
$482$	−764.018	−0.0721993
$483$	1079.03	0.101651
$484$	14655.4	1.37635
$485$	1510.41	0.141410
$486$	−8639.31	−0.806352
$487$	−2872.91	−0.267318	−0.133659	−	0.991027i	$-0.542673\pi$
$487$	−2872.91	−0.267318	−0.133659	+	0.991027i	$0.542673\pi$
$488$	53.5390	0.00496638
$489$	701.633	0.0648854
$490$	6308.27	0.581589
$491$	9727.08	0.894046	0.447023	−	0.894522i	$-0.352484\pi$
$491$	9727.08	0.894046	0.447023	+	0.894522i	$0.352484\pi$
$492$	1223.05	0.112072
$493$	468.988	0.0428441
$494$	−12252.0	−1.11588
$495$	−6826.64	−0.619868
$496$	−167.461	−0.0151597
$497$	−4920.78	−0.444119
$498$	−343.556	−0.0309139
$499$	10723.6	0.962032	0.481016	−	0.876712i	$-0.340268\pi$
$499$	10723.6	0.962032	0.481016	+	0.876712i	$0.340268\pi$
$500$	−1297.15	−0.116020
$501$	870.934	0.0776656
$502$	3.91116	0.000347736 0
$503$	1896.04	0.168072	0.0840359	−	0.996463i	$-0.473219\pi$
$503$	1896.04	0.168072	0.0840359	+	0.996463i	$0.473219\pi$
$504$	−1853.67	−0.163828
$505$	4757.22	0.419195
$506$	35957.9	3.15914
$507$	1408.14	0.123349
$508$	−23719.2	−2.07160
$509$	−11790.8	−1.02675	−0.513377	−	0.858163i	$-0.671606\pi$
$509$	−11790.8	−1.02675	−0.513377	+	0.858163i	$0.671606\pi$
$510$	−102.619	−0.00890991
$511$	4715.40	0.408214
$512$	13044.7	1.12598
$513$	5388.43	0.463752
$514$	7401.18	0.635121
$515$	−4849.98	−0.414982
$516$	1115.50	0.0951688
$517$	−12499.1	−1.06327
$518$	11851.4	1.00525
$519$	−957.389	−0.0809725
$520$	1384.79	0.116783
$521$	11954.0	1.00521	0.502606	−	0.864515i	$-0.332374\pi$
$521$	11954.0	1.00521	0.502606	+	0.864515i	$0.332374\pi$
$522$	10569.7	0.886255
$523$	−3495.14	−0.292221	−0.146111	−	0.989268i	$-0.546676\pi$
$523$	−3495.14	−0.292221	−0.146111	+	0.989268i	$0.546676\pi$
$524$	−13844.3	−1.15418
$525$	−168.442	−0.0140027
$526$	−12103.3	−1.00328
$527$	−21.1111	−0.00174500
$528$	−1989.10	−0.163948
$529$	13480.4	1.10794
$530$	4011.53	0.328773
$531$	6037.86	0.493448
$532$	7614.86	0.620575
$533$	3317.41	0.269593
$534$	511.285	0.0414334
$535$	1738.72	0.140507
$536$	−3424.58	−0.275969
$537$	2820.98	0.226693
$538$	−5013.65	−0.401773
$539$	15414.7	1.23183
$540$	−2658.64	−0.211870
$541$	8672.76	0.689226	0.344613	−	0.938745i	$-0.388010\pi$
$541$	8672.76	0.689226	0.344613	+	0.938745i	$0.388010\pi$
$542$	−24357.9	−1.93037
$543$	2651.76	0.209572
$544$	1240.26	0.0977494
$545$	−1707.55	−0.134208
$546$	784.993	0.0615286
$547$	−444.731	−0.0347629	−0.0173815	−	0.999849i	$-0.505533\pi$
$547$	−444.731	−0.0347629	−0.0173815	+	0.999849i	$0.505533\pi$
$548$	10348.7	0.806706
$549$	−136.953	−0.0106467
$550$	−5613.24	−0.435181
$551$	−9946.60	−0.769037
$552$	1575.80	0.121504
$553$	3576.83	0.275049
$554$	11687.2	0.896284
$555$	1912.71	0.146288
$556$	24542.8	1.87203
$557$	−9207.39	−0.700412	−0.350206	−	0.936673i	$-0.613888\pi$
$557$	−9207.39	−0.700412	−0.350206	+	0.936673i	$0.613888\pi$
$558$	−475.787	−0.0360962
$559$	3025.70	0.228933
$560$	1372.28	0.103553
$561$	−250.757	−0.0188716
$562$	36533.2	2.74210
$563$	−5847.84	−0.437757	−0.218878	−	0.975752i	$-0.570240\pi$
$563$	−5847.84	−0.437757	−0.218878	+	0.975752i	$0.570240\pi$
$564$	−2391.13	−0.178519
$565$	−8506.11	−0.633372
$566$	17760.7	1.31897
$567$	4566.07	0.338196
$568$	−7186.24	−0.530859
$569$	20778.2	1.53088	0.765439	−	0.643509i	$-0.222522\pi$
$569$	20778.2	1.53088	0.765439	+	0.643509i	$0.222522\pi$
$570$	2176.41	0.159930
$571$	13434.1	0.984588	0.492294	−	0.870429i	$-0.336158\pi$
$571$	13434.1	0.984588	0.492294	+	0.870429i	$0.336158\pi$
$572$	14771.6	1.07978
$573$	−1859.75	−0.135588
$574$	−3651.36	−0.265514
$575$	−4003.70	−0.290375
$576$	19749.8	1.42866
$577$	−4042.64	−0.291677	−0.145838	−	0.989308i	$-0.546588\pi$
$577$	−4042.64	−0.291677	−0.145838	+	0.989308i	$0.546588\pi$
$578$	−20955.9	−1.50805
$579$	1581.60	0.113522
$580$	4907.63	0.351342
$581$	579.174	0.0413566
$582$	1250.38	0.0890551
$583$	9802.46	0.696357
$584$	6886.32	0.487942
$585$	−3542.32	−0.250354
$586$	38233.1	2.69522
$587$	18469.8	1.29869	0.649343	−	0.760496i	$-0.275044\pi$
$587$	18469.8	1.29869	0.649343	+	0.760496i	$0.275044\pi$
$588$	2948.90	0.206821
$589$	447.737	0.0313220
$590$	4964.66	0.346427
$591$	5197.44	0.361749
$592$	−15582.7	−1.08183
$593$	−27615.5	−1.91237	−0.956183	−	0.292770i	$-0.905423\pi$
$593$	−27615.5	−1.91237	−0.956183	+	0.292770i	$0.905423\pi$
$594$	−11504.9	−0.794702
$595$	172.998	0.0119197
$596$	378.267	0.0259974
$597$	1015.27	0.0696016
$598$	18658.5	1.27592
$599$	−14244.0	−0.971608	−0.485804	−	0.874068i	$-0.661473\pi$
$599$	−14244.0	−0.971608	−0.485804	+	0.874068i	$0.661473\pi$
$600$	−245.991	−0.0167376
$601$	−17520.0	−1.18911	−0.594557	−	0.804054i	$-0.702672\pi$
$601$	−17520.0	−1.18911	−0.594557	+	0.804054i	$0.702672\pi$
$602$	−3330.28	−0.225469
$603$	8760.14	0.591609
$604$	26226.2	1.76677
$605$	−7061.35	−0.474520
$606$	3938.24	0.263994
$607$	−3521.60	−0.235482	−0.117741	−	0.993044i	$-0.537565\pi$
$607$	−3521.60	−0.235482	−0.117741	+	0.993044i	$0.537565\pi$
$608$	−26304.2	−1.75457
$609$	637.285	0.0424041
$610$	−112.611	−0.00747455
$611$	−6485.74	−0.429435
$612$	1341.28	0.0885918
$613$	−7234.78	−0.476689	−0.238344	−	0.971181i	$-0.576605\pi$
$613$	−7234.78	−0.476689	−0.238344	+	0.971181i	$0.576605\pi$
$614$	−21415.5	−1.40759
$615$	−589.297	−0.0386386
$616$	−3724.47	−0.243609
$617$	−4347.40	−0.283662	−0.141831	−	0.989891i	$-0.545299\pi$
$617$	−4347.40	−0.283662	−0.141831	+	0.989891i	$0.545299\pi$
$618$	−4015.03	−0.261340
$619$	30075.4	1.95288	0.976440	−	0.215788i	$-0.0692321\pi$
$619$	30075.4	1.95288	0.976440	+	0.215788i	$0.0692321\pi$
$620$	−220.912	−0.0143098
$621$	−8206.00	−0.530266
$622$	−34102.1	−2.19834
$623$	−861.935	−0.0554297
$624$	−1032.14	−0.0662157
$625$	625.000	0.0400000
$626$	−34567.1	−2.20700
$627$	5318.22	0.338739
$628$	8090.25	0.514070
$629$	−1964.44	−0.124527
$630$	3898.91	0.246565
$631$	10308.2	0.650337	0.325169	−	0.945656i	$-0.394579\pi$
$631$	10308.2	0.650337	0.325169	+	0.945656i	$0.394579\pi$
$632$	5223.56	0.328769
$633$	1165.73	0.0731966
$634$	−27150.5	−1.70076
$635$	11428.6	0.714219
$636$	1875.25	0.116916
$637$	7998.65	0.497516
$638$	21237.2	1.31785
$639$	18382.5	1.13803
$640$	6234.05	0.385035
$641$	29151.5	1.79628	0.898139	−	0.439712i	$-0.144919\pi$
$641$	29151.5	1.79628	0.898139	+	0.439712i	$0.144919\pi$
$642$	1439.39	0.0884864
$643$	17612.1	1.08018	0.540088	−	0.841609i	$-0.318391\pi$
$643$	17612.1	1.08018	0.540088	+	0.841609i	$0.318391\pi$
$644$	−11596.6	−0.709582
$645$	−537.478	−0.0328111
$646$	−2235.27	−0.136139
$647$	16602.2	1.00881	0.504403	−	0.863468i	$-0.331712\pi$
$647$	16602.2	1.00881	0.504403	+	0.863468i	$0.331712\pi$
$648$	6668.24	0.404249
$649$	12131.5	0.733750
$650$	−2912.69	−0.175762
$651$	−28.6868	−0.00172707
$652$	−7540.65	−0.452937
$653$	10158.9	0.608804	0.304402	−	0.952544i	$-0.401543\pi$
$653$	10158.9	0.608804	0.304402	+	0.952544i	$0.401543\pi$
$654$	−1413.59	−0.0845192
$655$	6670.55	0.397923
$656$	4800.95	0.285740
$657$	−17615.3	−1.04603
$658$	7138.63	0.422937
$659$	11885.5	0.702570	0.351285	−	0.936269i	$-0.385745\pi$
$659$	11885.5	0.702570	0.351285	+	0.936269i	$0.385745\pi$
$660$	−2624.00	−0.154756
$661$	−4255.12	−0.250386	−0.125193	−	0.992132i	$-0.539955\pi$
$661$	−4255.12	−0.250386	−0.125193	+	0.992132i	$0.539955\pi$
$662$	23665.2	1.38939
$663$	−130.117	−0.00762191
$664$	845.819	0.0494340
$665$	−3669.04	−0.213954
$666$	−44273.2	−2.57590
$667$	15147.6	0.879337
$668$	−9360.18	−0.542150
$669$	2316.24	0.133858
$670$	7203.07	0.415342
$671$	−275.172	−0.0158315
$672$	1685.33	0.0967454
$673$	10526.8	0.602941	0.301471	−	0.953475i	$-0.402522\pi$
$673$	10526.8	0.602941	0.301471	+	0.953475i	$0.402522\pi$
$674$	−33807.8	−1.93209
$675$	1281.00	0.0730457
$676$	−15133.7	−0.861043
$677$	5964.60	0.338609	0.169304	−	0.985564i	$-0.445848\pi$
$677$	5964.60	0.338609	0.169304	+	0.985564i	$0.445848\pi$
$678$	−7041.75	−0.398874
$679$	−2107.93	−0.119138
$680$	252.644	0.0142477
$681$	2861.47	0.161016
$682$	−955.971	−0.0536745
$683$	−8544.58	−0.478696	−0.239348	−	0.970934i	$-0.576934\pi$
$683$	−8544.58	−0.478696	−0.239348	+	0.970934i	$0.576934\pi$
$684$	−28446.8	−1.59019
$685$	−4986.29	−0.278126
$686$	−19064.2	−1.06104
$687$	2024.83	0.112448
$688$	4378.78	0.242645
$689$	5086.46	0.281246
$690$	−3314.44	−0.182868
$691$	−11730.3	−0.645793	−0.322896	−	0.946434i	$-0.604657\pi$
$691$	−11730.3	−0.645793	−0.322896	+	0.946434i	$0.604657\pi$
$692$	10289.3	0.565234
$693$	9527.26	0.522238
$694$	40849.5	2.23433
$695$	−11825.4	−0.645414
$696$	930.684	0.0506860
$697$	605.234	0.0328908
$698$	5186.05	0.281225
$699$	993.891	0.0537803
$700$	1810.30	0.0977469
$701$	−26902.9	−1.44951	−0.724756	−	0.689005i	$-0.758048\pi$
$701$	−26902.9	−1.44951	−0.724756	+	0.689005i	$0.758048\pi$
$702$	−5969.87	−0.320966
$703$	41663.0	2.23521
$704$	39682.2	2.12440
$705$	1152.11	0.0615475
$706$	−47852.9	−2.55095
$707$	−6639.18	−0.353171
$708$	2320.81	0.123194
$709$	27066.5	1.43371	0.716857	−	0.697220i	$-0.245580\pi$
$709$	27066.5	1.43371	0.716857	+	0.697220i	$0.245580\pi$
$710$	15115.1	0.798959
$711$	−13362.0	−0.704799
$712$	−1258.76	−0.0662556
$713$	−681.856	−0.0358144
$714$	143.215	0.00750658
$715$	−7117.37	−0.372272
$716$	−30317.9	−1.58245
$717$	−3941.93	−0.205320
$718$	22428.8	1.16579
$719$	−28978.7	−1.50309	−0.751547	−	0.659680i	$-0.770692\pi$
$719$	−28978.7	−1.50309	−0.751547	+	0.659680i	$0.770692\pi$
$720$	−5126.43	−0.265349
$721$	6768.63	0.349621
$722$	18003.5	0.928005
$723$	172.086	0.00885191
$724$	−28499.2	−1.46293
$725$	−2364.63	−0.121131
$726$	−5845.71	−0.298836
$727$	−489.486	−0.0249712	−0.0124856	−	0.999922i	$-0.503974\pi$
$727$	−489.486	−0.0249712	−0.0124856	+	0.999922i	$0.503974\pi$
$728$	−1932.62	−0.0983895
$729$	−15721.6	−0.798741
$730$	−14484.3	−0.734366
$731$	552.013	0.0279302
$732$	−52.6416	−0.00265805
$733$	33466.1	1.68636	0.843178	−	0.537634i	$-0.180682\pi$
$733$	33466.1	1.68636	0.843178	+	0.537634i	$0.180682\pi$
$734$	−9715.19	−0.488548
$735$	−1420.86	−0.0713051
$736$	40058.5	2.00622
$737$	17601.2	0.879714
$738$	13640.4	0.680365
$739$	35051.3	1.74477	0.872384	−	0.488821i	$-0.162573\pi$
$739$	35051.3	1.74477	0.872384	+	0.488821i	$0.162573\pi$
$740$	−20556.4	−1.02118
$741$	2759.61	0.136811
$742$	−5598.49	−0.276991
$743$	−5545.26	−0.273803	−0.136902	−	0.990585i	$-0.543714\pi$
$743$	−5545.26	−0.273803	−0.136902	+	0.990585i	$0.543714\pi$
$744$	−41.8939	−0.00206439
$745$	−182.259	−0.00896304
$746$	−60094.5	−2.94935
$747$	−2163.62	−0.105974
$748$	2694.96	0.131735
$749$	−2426.56	−0.118377
$750$	517.403	0.0251905
$751$	24128.8	1.17240	0.586200	−	0.810167i	$-0.300623\pi$
$751$	24128.8	1.17240	0.586200	+	0.810167i	$0.300623\pi$
$752$	−9386.14	−0.455156
$753$	−0.880939	−4.26337e−5 0
$754$	11019.9	0.532256
$755$	−12636.5	−0.609125
$756$	3710.40	0.178500
$757$	−9476.21	−0.454978	−0.227489	−	0.973781i	$-0.573052\pi$
$757$	−9476.21	−0.454978	−0.227489	+	0.973781i	$0.573052\pi$
$758$	−18270.0	−0.875455
$759$	−8099.08	−0.387323
$760$	−5358.23	−0.255741
$761$	−9722.18	−0.463113	−0.231556	−	0.972821i	$-0.574382\pi$
$761$	−9722.18	−0.463113	−0.231556	+	0.972821i	$0.574382\pi$
$762$	9461.09	0.449789
$763$	2383.05	0.113070
$764$	19987.3	0.946484
$765$	−646.267	−0.0305436
$766$	5713.01	0.269477
$767$	6295.00	0.296349
$768$	−691.533	−0.0324916
$769$	−13355.4	−0.626278	−0.313139	−	0.949707i	$-0.601381\pi$
$769$	−13355.4	−0.626278	−0.313139	+	0.949707i	$0.601381\pi$
$770$	7833.84	0.366639
$771$	−1667.02	−0.0778682
$772$	−16997.9	−0.792448
$773$	2832.38	0.131790	0.0658949	−	0.997827i	$-0.479010\pi$
$773$	2832.38	0.131790	0.0658949	+	0.997827i	$0.479010\pi$
$774$	12440.9	0.577751
$775$	106.442	0.00493354
$776$	−3078.39	−0.142407
$777$	−2669.38	−0.123248
$778$	51445.1	2.37069
$779$	−12836.2	−0.590378
$780$	−1361.58	−0.0625032
$781$	36934.9	1.69223
$782$	3404.08	0.155665
$783$	−4846.55	−0.221203
$784$	11575.6	0.527315
$785$	−3898.10	−0.177235
$786$	5522.18	0.250598
$787$	−20653.8	−0.935485	−0.467743	−	0.883865i	$-0.654933\pi$
$787$	−20653.8	−0.935485	−0.467743	+	0.883865i	$0.654933\pi$
$788$	−55858.3	−2.52522
$789$	2726.11	0.123006
$790$	−10986.9	−0.494807
$791$	11871.1	0.533615
$792$	13913.5	0.624236
$793$	−142.786	−0.00639405
$794$	−60127.4	−2.68746
$795$	−903.547	−0.0403088
$796$	−10911.4	−0.485859
$797$	19838.2	0.881689	0.440845	−	0.897583i	$-0.354679\pi$
$797$	19838.2	0.881689	0.440845	+	0.897583i	$0.354679\pi$
$798$	−3037.40	−0.134740
$799$	−1183.27	−0.0523918
$800$	−6253.36	−0.276362
$801$	3219.93	0.142036
$802$	−9367.67	−0.412449
$803$	−35393.4	−1.55542
$804$	3367.19	0.147701
$805$	5587.56	0.244641
$806$	−496.050	−0.0216782
$807$	1129.26	0.0492589
$808$	−9695.77	−0.422149
$809$	268.004	0.0116471	0.00582356	−	0.999983i	$-0.498146\pi$
$809$	268.004	0.0116471	0.00582356	+	0.999983i	$0.498146\pi$
$810$	−14025.6	−0.608406
$811$	43842.9	1.89831	0.949157	−	0.314802i	$-0.101938\pi$
$811$	43842.9	1.89831	0.949157	+	0.314802i	$0.101938\pi$
$812$	−6849.09	−0.296005
$813$	5486.31	0.236671
$814$	−88955.5	−3.83033
$815$	3633.29	0.156158
$816$	−188.305	−0.00807843
$817$	−11707.4	−0.501336
$818$	−51881.1	−2.21758
$819$	4943.66	0.210923
$820$	6333.35	0.269720
$821$	8176.44	0.347576	0.173788	−	0.984783i	$-0.444399\pi$
$821$	8176.44	0.347576	0.173788	+	0.984783i	$0.444399\pi$
$822$	−4127.87	−0.175153
$823$	30938.9	1.31041	0.655203	−	0.755453i	$-0.272583\pi$
$823$	30938.9	1.31041	0.655203	+	0.755453i	$0.272583\pi$
$824$	9884.83	0.417906
$825$	1264.31	0.0533548
$826$	−6928.69	−0.291864
$827$	−24349.4	−1.02383	−0.511917	−	0.859035i	$-0.671065\pi$
$827$	−24349.4	−1.02383	−0.511917	+	0.859035i	$0.671065\pi$
$828$	43321.4	1.81827
$829$	37034.8	1.55159	0.775796	−	0.630983i	$-0.217348\pi$
$829$	37034.8	1.55159	0.775796	+	0.630983i	$0.217348\pi$
$830$	−1779.05	−0.0743995
$831$	−2632.40	−0.109888
$832$	20590.9	0.858008
$833$	1459.29	0.0606978
$834$	−9789.60	−0.406458
$835$	4509.99	0.186916
$836$	−57156.4	−2.36459
$837$	218.163	0.00900934
$838$	28076.7	1.15739
$839$	−26184.0	−1.07744	−0.538721	−	0.842484i	$-0.681092\pi$
$839$	−26184.0	−1.07744	−0.538721	+	0.842484i	$0.681092\pi$
$840$	343.305	0.0141014
$841$	−15442.7	−0.633181
$842$	31379.1	1.28432
$843$	−8228.66	−0.336192
$844$	−12528.4	−0.510954
$845$	7291.82	0.296859
$846$	−26667.7	−1.08375
$847$	9854.83	0.399783
$848$	7361.11	0.298092
$849$	−4000.38	−0.161711
$850$	−531.396	−0.0214432
$851$	−63448.4	−2.55580
$852$	7065.80	0.284120
$853$	−17761.7	−0.712952	−0.356476	−	0.934304i	$-0.616022\pi$
$853$	−17761.7	−0.712952	−0.356476	+	0.934304i	$0.616022\pi$
$854$	157.160	0.00629730
$855$	13706.4	0.548246
$856$	−3543.72	−0.141497
$857$	29199.5	1.16387	0.581934	−	0.813236i	$-0.302296\pi$
$857$	29199.5	1.16387	0.581934	+	0.813236i	$0.302296\pi$
$858$	−5892.08	−0.234443
$859$	29166.3	1.15849	0.579244	−	0.815155i	$-0.303348\pi$
$859$	29166.3	1.15849	0.579244	+	0.815155i	$0.303348\pi$
$860$	5776.43	0.229040
$861$	822.424	0.0325530
$862$	−68415.5	−2.70330
$863$	−20705.9	−0.816727	−0.408364	−	0.912819i	$-0.633900\pi$
$863$	−20705.9	−0.816727	−0.408364	+	0.912819i	$0.633900\pi$
$864$	−12816.9	−0.504676
$865$	−4957.68	−0.194874
$866$	−29971.4	−1.17606
$867$	4720.07	0.184893
$868$	308.305	0.0120560
$869$	−26847.4	−1.04803
$870$	−1957.55	−0.0762840
$871$	9133.21	0.355301
$872$	3480.18	0.135154
$873$	7874.57	0.305285
$874$	−72195.9	−2.79412
$875$	−872.250	−0.0336999
$876$	−6770.90	−0.261150
$877$	−22623.3	−0.871078	−0.435539	−	0.900170i	$-0.643442\pi$
$877$	−22623.3	−0.871078	−0.435539	+	0.900170i	$0.643442\pi$
$878$	73325.1	2.81845
$879$	−8611.53	−0.330444
$880$	−10300.2	−0.394569
$881$	−47624.8	−1.82125	−0.910625	−	0.413234i	$-0.864399\pi$
$881$	−47624.8	−1.82125	−0.910625	+	0.413234i	$0.864399\pi$
$882$	32888.4	1.25557
$883$	−7294.60	−0.278010	−0.139005	−	0.990292i	$-0.544390\pi$
$883$	−7294.60	−0.278010	−0.139005	+	0.990292i	$0.544390\pi$
$884$	1398.41	0.0532053
$885$	−1118.23	−0.0424733
$886$	−13733.5	−0.520751
$887$	21150.9	0.800652	0.400326	−	0.916373i	$-0.368897\pi$
$887$	21150.9	0.800652	0.400326	+	0.916373i	$0.368897\pi$
$888$	−3898.33	−0.147319
$889$	−15949.7	−0.601728
$890$	2647.60	0.0997167
$891$	−34272.5	−1.28863
$892$	−24893.3	−0.934404
$893$	25095.5	0.940413
$894$	−150.883	−0.00564460
$895$	14608.0	0.545576
$896$	−8700.24	−0.324391
$897$	−4202.59	−0.156433
$898$	−368.421	−0.0136908
$899$	−402.711	−0.0149401
$900$	−6762.72	−0.250471
$901$	927.982	0.0343125
$902$	27406.8	1.01169
$903$	750.104	0.0276433
$904$	17336.5	0.637835
$905$	13731.7	0.504372
$906$	−10461.1	−0.383605
$907$	−14494.5	−0.530630	−0.265315	−	0.964162i	$-0.585476\pi$
$907$	−14494.5	−0.530630	−0.265315	+	0.964162i	$0.585476\pi$
$908$	−30753.1	−1.12398
$909$	24801.9	0.904982
$910$	4064.95	0.148079
$911$	23395.2	0.850842	0.425421	−	0.904995i	$-0.360126\pi$
$911$	23395.2	0.850842	0.425421	+	0.904995i	$0.360126\pi$
$912$	3993.69	0.145005
$913$	−4347.23	−0.157582
$914$	−20351.6	−0.736512
$915$	25.3642	0.000916408 0
$916$	−21761.4	−0.784953
$917$	−9309.42	−0.335250
$918$	−1089.15	−0.0391584
$919$	18784.8	0.674271	0.337135	−	0.941456i	$-0.390542\pi$
$919$	18784.8	0.674271	0.337135	+	0.941456i	$0.390542\pi$
$920$	8160.01	0.292421
$921$	4823.57	0.172576
$922$	6242.60	0.222982
$923$	19165.4	0.683463
$924$	3662.05	0.130382
$925$	9904.65	0.352068
$926$	5948.22	0.211091
$927$	−25285.5	−0.895886
$928$	23659.0	0.836901
$929$	−41261.1	−1.45719	−0.728597	−	0.684943i	$-0.759827\pi$
$929$	−41261.1	−1.45719	−0.728597	+	0.684943i	$0.759827\pi$
$930$	88.1172	0.00310696
$931$	−30949.5	−1.08950
$932$	−10681.6	−0.375417
$933$	7681.07	0.269525
$934$	82004.8	2.87289
$935$	−1298.50	−0.0454178
$936$	7219.67	0.252118
$937$	29809.8	1.03932	0.519661	−	0.854372i	$-0.326058\pi$
$937$	29809.8	1.03932	0.519661	+	0.854372i	$0.326058\pi$
$938$	−10052.6	−0.349925
$939$	7785.82	0.270586
$940$	−12382.1	−0.429637
$941$	4889.81	0.169398	0.0846988	−	0.996407i	$-0.473007\pi$
$941$	4889.81	0.169398	0.0846988	+	0.996407i	$0.473007\pi$
$942$	−3227.03	−0.111616
$943$	19548.1	0.675053
$944$	9110.11	0.314098
$945$	−1787.77	−0.0615409
$946$	24996.8	0.859107
$947$	51182.2	1.75628	0.878141	−	0.478402i	$-0.158784\pi$
$947$	51182.2	1.75628	0.878141	+	0.478402i	$0.158784\pi$
$948$	−5136.02	−0.175960
$949$	−18365.5	−0.628208
$950$	11270.2	0.384898
$951$	6115.31	0.208520
$952$	−352.590	−0.0120037
$953$	−9309.13	−0.316424	−0.158212	−	0.987405i	$-0.550573\pi$
$953$	−9309.13	−0.316424	−0.158212	+	0.987405i	$0.550573\pi$
$954$	20914.3	0.709774
$955$	−9630.41	−0.326317
$956$	42365.1	1.43325
$957$	−4783.40	−0.161573
$958$	−35453.5	−1.19567
$959$	6958.86	0.234321
$960$	−3657.73	−0.122971
$961$	−29772.9	−0.999392
$962$	−46158.7	−1.54700
$963$	9064.88	0.303335
$964$	−1849.45	−0.0617913
$965$	8190.07	0.273210
$966$	4625.64	0.154066
$967$	−41654.5	−1.38523	−0.692615	−	0.721308i	$-0.743541\pi$
$967$	−41654.5	−1.38523	−0.692615	+	0.721308i	$0.743541\pi$
$968$	14391.9	0.477864
$969$	503.467	0.0166911
$970$	6474.91	0.214326
$971$	−34443.6	−1.13836	−0.569181	−	0.822212i	$-0.692740\pi$
$971$	−34443.6	−1.13836	−0.569181	+	0.822212i	$0.692740\pi$
$972$	−20913.1	−0.690112
$973$	16503.5	0.543760
$974$	−12315.8	−0.405156
$975$	656.047	0.0215491
$976$	−206.639	−0.00677702
$977$	−32195.4	−1.05427	−0.527134	−	0.849782i	$-0.676734\pi$
$977$	−32195.4	−1.05427	−0.527134	+	0.849782i	$0.676734\pi$
$978$	3007.80	0.0983425
$979$	6469.61	0.211205
$980$	15270.4	0.497750
$981$	−8902.36	−0.289736
$982$	41698.6	1.35505
$983$	55304.6	1.79445	0.897224	−	0.441576i	$-0.145580\pi$
$983$	55304.6	1.79445	0.897224	+	0.441576i	$0.145580\pi$
$984$	1201.06	0.0389109
$985$	26914.1	0.870612
$986$	2010.49	0.0649360
$987$	−1607.89	−0.0518537
$988$	−29658.3	−0.955016
$989$	17829.2	0.573241
$990$	−29264.8	−0.939493
$991$	−44112.6	−1.41401	−0.707005	−	0.707209i	$-0.749954\pi$
$991$	−44112.6	−1.41401	−0.707005	+	0.707209i	$0.749954\pi$
$992$	−1064.99	−0.0340861
$993$	−5330.30	−0.170344
$994$	−21094.7	−0.673122
$995$	5257.40	0.167508
$996$	−831.643	−0.0264575
$997$	−23829.2	−0.756948	−0.378474	−	0.925612i	$-0.623551\pi$
$997$	−23829.2	−0.756948	−0.378474	+	0.925612i	$0.623551\pi$
$998$	45970.6	1.45809
$999$	20300.6	0.642926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.4.a.c.1.19	✓	21
5.4	even	2		2075.4.a.g.1.3		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.4.a.c.1.19	✓	21	1.1	even	1	trivial
2075.4.a.g.1.3		21	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	415.a (trivial)

\(f(q)\)	\(=\)	\(q+4.28686 q^{2} -0.965562 q^{3} +10.3772 q^{4} -5.00000 q^{5} -4.13923 q^{6} +6.97800 q^{7} +10.1906 q^{8} -26.0677 q^{9} -21.4343 q^{10} -52.3762 q^{11} -10.0198 q^{12} -27.1779 q^{13} +29.9137 q^{14} +4.82781 q^{15} -39.3317 q^{16} -4.95837 q^{17} -111.749 q^{18} +105.160 q^{19} -51.8858 q^{20} -6.73769 q^{21} -224.530 q^{22} -160.148 q^{23} -9.83965 q^{24} +25.0000 q^{25} -116.508 q^{26} +51.2401 q^{27} +72.4119 q^{28} -94.5851 q^{29} +20.6961 q^{30} +4.25766 q^{31} -250.134 q^{32} +50.5725 q^{33} -21.2558 q^{34} -34.8900 q^{35} -270.509 q^{36} +396.186 q^{37} +450.807 q^{38} +26.2419 q^{39} -50.9530 q^{40} -122.063 q^{41} -28.8835 q^{42} -111.330 q^{43} -543.517 q^{44} +130.338 q^{45} -686.532 q^{46} +238.641 q^{47} +37.9772 q^{48} -294.307 q^{49} +107.172 q^{50} +4.78761 q^{51} -282.029 q^{52} -187.155 q^{53} +219.659 q^{54} +261.881 q^{55} +71.1100 q^{56} -101.539 q^{57} -405.473 q^{58} -231.622 q^{59} +50.0990 q^{60} +5.25376 q^{61} +18.2520 q^{62} -181.900 q^{63} -757.637 q^{64} +135.889 q^{65} +216.797 q^{66} -336.053 q^{67} -51.4539 q^{68} +154.633 q^{69} -149.569 q^{70} -705.184 q^{71} -265.645 q^{72} +675.753 q^{73} +1698.39 q^{74} -24.1390 q^{75} +1091.27 q^{76} -365.482 q^{77} +112.495 q^{78} +512.587 q^{79} +196.659 q^{80} +654.352 q^{81} -523.267 q^{82} +83.0000 q^{83} -69.9182 q^{84} +24.7919 q^{85} -477.254 q^{86} +91.3277 q^{87} -533.745 q^{88} -123.522 q^{89} +558.743 q^{90} -189.647 q^{91} -1661.88 q^{92} -4.11103 q^{93} +1023.02 q^{94} -525.801 q^{95} +241.520 q^{96} -302.081 q^{97} -1261.65 q^{98} +1365.33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 5 q^{2} - 12 q^{3} + 87 q^{4} - 105 q^{5} - 7 q^{6} - 11 q^{7} - 84 q^{8} + 153 q^{9} + 25 q^{10} - 30 q^{11} - 244 q^{12} - 89 q^{13} - 191 q^{14} + 60 q^{15} + 583 q^{16} - 357 q^{17} - 281 q^{18}+ \cdots - 5369 q^{99}+O(q^{100}) \) 21 * q - 5 * q^2 - 12 * q^3 + 87 * q^4 - 105 * q^5 - 7 * q^6 - 11 * q^7 - 84 * q^8 + 153 * q^9 + 25 * q^10 - 30 * q^11 - 244 * q^12 - 89 * q^13 - 191 * q^14 + 60 * q^15 + 583 * q^16 - 357 * q^17 - 281 * q^18 - 175 * q^19 - 435 * q^20 - 41 * q^21 - 122 * q^22 - 554 * q^23 - 136 * q^24 + 525 * q^25 - 933 * q^26 - 555 * q^27 - 773 * q^28 - 96 * q^29 + 35 * q^30 + 99 * q^31 - 1307 * q^32 - 1097 * q^33 + 393 * q^34 + 55 * q^35 - 746 * q^36 - 756 * q^37 - 1192 * q^38 + 40 * q^39 + 420 * q^40 - 1436 * q^41 - 1112 * q^42 - 1003 * q^43 - 706 * q^44 - 765 * q^45 - 775 * q^46 - 858 * q^47 - 2837 * q^48 + 1384 * q^49 - 125 * q^50 - 279 * q^51 - 1092 * q^52 - 1625 * q^53 - 269 * q^54 + 150 * q^55 - 3806 * q^56 - 1740 * q^57 - 1087 * q^58 - 1224 * q^59 + 1220 * q^60 - 794 * q^61 - 2979 * q^62 - 1236 * q^63 + 2570 * q^64 + 445 * q^65 - 1094 * q^66 - 571 * q^67 - 4064 * q^68 + 774 * q^69 + 955 * q^70 - 1078 * q^71 - 2610 * q^72 - 1986 * q^73 - 1872 * q^74 - 300 * q^75 - 2854 * q^76 - 2519 * q^77 - 2449 * q^78 - 432 * q^79 - 2915 * q^80 - 219 * q^81 - 2143 * q^82 + 1743 * q^83 - 377 * q^84 + 1785 * q^85 + 1556 * q^86 - 5681 * q^87 - 3158 * q^88 - 6252 * q^89 + 1405 * q^90 + 684 * q^91 - 7623 * q^92 - 3314 * q^93 - 3877 * q^94 + 875 * q^95 - 954 * q^96 - 4880 * q^97 - 5072 * q^98 - 5369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more